Theorem setinds2f

Description: _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	setinds2f.1	⊢ Ⅎ 𝑥 𝜓
	setinds2f.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	setinds2f.3	⊢ ( ∀ 𝑦 ∈ 𝑥 𝜓 → 𝜑 )
Assertion	setinds2f	⊢ 𝜑

Proof

Step	Hyp	Ref	Expression
1		setinds2f.1	⊢ Ⅎ 𝑥 𝜓
2		setinds2f.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		setinds2f.3	⊢ ( ∀ 𝑦 ∈ 𝑥 𝜓 → 𝜑 )
4		sbsbc	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
5	1 2	sbiev	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
6	4 5	bitr3i	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
7	6	ralbii	⊢ ( ∀ 𝑦 ∈ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ∈ 𝑥 𝜓 )
8	7 3	sylbi	⊢ ( ∀ 𝑦 ∈ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 )
9	8	setinds	⊢ 𝜑